Anisotropic phonon coupling in the relaxor ferroelectric $\left({\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}\right){\mathrm{TiO}}_3$ near its high-temperature phase transition

The lead free relaxor ${\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}{\mathrm{TiO}}_3$ (NBT) undergoes a structural cubic-to-tetragonal transition near 800 K which is caused by the cooperative rotations of ${\mathrm{O}}_6$ octahedra. These rotations are also accompanied by the displacements of the cations and the formation of the polar nanodomains (PNDs) that are responsible for the characteristic dielectric dispersion of relaxor ferroelectrics. Because of their intrinsic properties, spontaneous polarization, and lack of inversion symmetry, these PNDs are also piezoelectric and can mediate an interaction between polarization and strain or couple the optic and acoustic phonons. Because PNDs introduce a local tetragonal symmetry, the phonon coupling they mediate is found to be anisotropic. In this paper we present inelastic neutron scattering results on coupled transverse acoustic (TA) and transverse optic (TO) phonons in the [110] and [001] directions and across the cubic-tetragonal phase transition at $T_C\sim 800$ K. The phonon spectra are analyzed using a mode coupling model. In the [110] direction, as in other relaxors and some ferroelectric perovskites, a precipitous drop of the TO phonon into the TA branch or “waterfall” is observed at a certain $q_{\mathrm{wf}}\sim 0.14$ r.l.u. In the [001] direction, the highly overdamped line shape can be fitted with closely positioned bare mode energies which are largely overlapping along the dispersion curves. Two competing lattice coupling mechanism are proposed to explain these observations.

Figure 1

$\theta \text{−}2\theta$ scan at (002) Bragg reflection measured across $T_C=800$ K. The inset shows the sharp increase of integrated intensity at $T_C=800$ K.

Figure 2

Constant $q$ scans near (002) zone at (a) $\Gamma$ point, (b) (0.05, 0.05, 2), and (c) (0.5, 0.5, 2). The inset in (b) indicates that the central peak background intensity is very high at (0.05, 0.05, 2). The spectra in (c) are offset by a constant for clarity and the dashed line is guide to the eye for the average peak position.

Figure 3

Near zone center plots of the [110] phonon fitted with mode coupling equation at (a) $\xi =0.1$ and (c) $\xi =0.15$. The resulting coupling constants are plotted in (b) and (d), respectively.

Figure 4

(a) and (b) Constant $q$ scans of the [110] phonon at 723 K. The magenta lines are the fitted result as described in the text. We also show a set of constant $E$ scans in (c). The dispersion relations in (d) are obtained from fitting to both constant $Q$ and constant $E$ scans. The vertical bars represent the FWHM of the peak. The solid lines in (d) are guides for the eye.

Figure 5

Temperature overplots of constant $q$ scans at (a) near zone center (2, 2, 0.1) and (b) near zone boundary (2, 2, 0.4). The spectra between temperatures are offset by a constant for clarity.

Figure 6

(a) (2, 2, 0) and (b) (2, 2, 0.1) constant $q$ scans plotted in log scale. (c) (2, 2, $q)$ phonons measured in constant $q$ mode at 873 K and fitted with the MC model. Phonon dispersion relation obtained by assuming real coupling $\left({\Delta }_{12}\right)$ and imaginary coupling $\left({\Gamma }_{12}\right)$ are shown in (d) and (e), respectively. The inset in (d) shows the real coupling constant ${\Delta }_{12}$ as function of $q$. The solid gray circles are apparent spectral intensities at (2, 2, 0) and (2, 2, 0.1). The dashed lines are guides for the eye, see text for explanation.

Figure 7

Data and fitted curve of Eq. (2) for constant $q$ scan at (a) (0.05, 0.05, 2) and (b) (0.1, 0.1, 2). The fitting parameters are shown in Table 2.